Effect of stars on the dark matter spike around a black hole: A tale of two treatments

Stuart L. Shapiro and Douglas C. Heggie

Phys. Rev. D 106, 043018 – Published 17 August 2022

Abstract

We revisit the role that gravitational scattering off stars plays in establishing the steady-state distribution of collisionless dark matter (DM) around a massive black hole (BH). This is a physically interesting problem that has potentially observable signatures, such as $γ$ rays from DM annihilation in a density spike. The system serves as a laboratory for comparing two different dynamical approaches, both of which have been widely used: a Fokker-Planck treatment and a two-component conduction fluid treatment. In our Fokker-Planck analysis we extend a previous analytic model to account for a nonzero flux of DM particles into the BH, as well as a cutoff in the distribution function near the BH due to relativistic effects or, further out, possible DM annihilation. In our two-fluid analysis, following an approximate analytic treatment, we recast the equations as a “heated Bondi accretion” problem and solve the equations numerically without approximation. While both the Fokker-Planck and two-fluid methods yield basically the same DM density and velocity dispersion profiles away from the boundaries in the spike interior, there are other differences, especially the determination of the DM accretion rate. We discuss limitations of the two treatments, including the assumption of an isotropic velocity dispersion.

Received 29 June 2022
Accepted 8 August 2022

DOI:https://doi.org/10.1103/PhysRevD.106.043018

Physics Subject Headings (PhySH)

Particle dark matter

Classical black holes

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Stuart L. Shapiro ^1,* and Douglas C. Heggie ²

¹Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
²School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edingburgh, Kings Buildings, Edinburg EH9 3FD, United Kingdom

^*Also at Department of Astronomy and NCSA, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA.

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Issue

Vol. 106, Iss. 4 — 15 August 2022

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Images

Figure 1
Fokker-Planck solution for the DM steady-state density profile $ρ (r)$ in the spike around a massive black hole, allowing for background stars. The DM distribution function cuts off at $r_{cut} = r_{mb}$ (upper figure) and $r_{ann}$ (lower figure); vertical arrows show the location of $r_{cut}$ . Three stellar density profiles $ρ_{*} \sim r^{- β}$ are chosen for each figure: $β = 1$ (solid, red); $7 / 4$ (dotted, blue); 1.4 (dashed, green). The three curves are nearly indistinguishable in the plot. The densities and radii are normalized to their values near the spike outer boundary at $r_{h}$ . Parameters are chosen that characterize a spike around Sgr A* in the Galactic Center (see text).
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Figure 2
Fokker-Planck solution for the DM steady-state velocity dispersion profile in the spike around a massive black hole, allowing for background stars. Results are plotted for the cases shown in Fig. 1 and the labeling is the same as in that figure. The velocity dispersion is normalized to the square of the local circular velocity $M_{bh} / r$ .
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Figure 3
Two-component fluid solution for the DM steady-state density profile $ρ (r)$ in the spike around a massive black hole, allowing for background stars. The stellar density is given by $ρ_{*} \sim r^{- β}$ , with $β = 1.4$ . Four accretion rates are chosen according to Eq. (43), with $q = 0.1$ (solid, red); 1 (dotted, blue); 10 (dashed, green); 100 (dot-dashed, magenta). The densities and radii are normalized to their values at the spike outer boundary at $r_{h}$ . Parameters are chosen that characterize a spike around Sgr A* in the Galactic Center (see text).
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Figure 4
Two-component fluid solution for the DM steady-state dimensionless heating parameter $R (r)$ in the spike around a massive black hole, allowing for background stars [see Eq. (29)]. Results are plotted for the cases shown in Fig. 3 and the labeling is the same as in that figure.
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Figure 5
Two-component fluid solution for the DM steady-state velocity dispersion profile $v (r)$ in the spike around a massive black hole, allowing for background stars. Results are plotted for the cases shown in Fig. 3 and the labeling is the same as in that figure. The velocity dispersion is normalized to the square of the local circular velocity $M_{bh} / r$ .
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Figure 6
Two-component fluid solution for the DM steady-state Mach number $u (r) / a (r)$ in the spike around a massive black hole, allowing for background stars. Results are plotted for the cases shown in Fig. 3 and the labeling is the same as in that figure.
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